Section 2.5 Critical Numbers – Relative Maximum and Minimum Points

If f ‘ (a) > 0 The graph of f(x) is INCREASING at x = a x = a

If f ‘ (a) < 0 The graph of f(x) is DECREASING at x = a x = a

This is the graph of g(x) This is the graph of f(x) If f ‘ (a) = 0, a maximum or minimum MAY exist.

This is the graph of g ‘ (x) This is the graph of f ‘ (x)

A change from increasing to decreasing indicates a maximum

A change from decreasing to increasing indicates a minimum

If f ‘ (x) > 0 on an interval (a, b), f is increasing on (a, b). If f ‘ (x) < 0 on an interval (a, b), f is decreasing on (a, b). If f ‘ (c) = 0 or f ‘ (c) does not exist, c is a critical number If f ‘ (c) = 0, a relative maximum will exist IF f ‘ (x) changes from positive to negative. If f ‘ (c) = 0, a relative minimum will exist IF f ‘ (x) changes from negative to positive. A RELATIVE max/min is a high/low point around the area. An ABSOLUTE max/min is THE high/low point on an interval. FACTS ABOUT f ‘ (x) = 0

A.Where are the relative extrema of f? B.On what interval(s) is f ‘ < 0? (1, 3) C.On what interval(s) is f ‘ > 0? (-1, 1) and (3, 5) D. Where are the zero(s) of f? x = 0 This is the graph of f(x) on the interval [-1, 5]. x = -1, x = 1, x = 3, x = 5

A.Where are the relative extrema of f? B. On what interval(s) is f ‘ < 0? [-1, 0) C. On what interval(s) is f ‘ > 0? (0, 5] D.On what interval(s) is f “ > 0? (-1, 1), (3, 5) This is the graph of f ‘ (x) on the interval [-1, 5]. x = -1, x = 0, x = 5

A.Where are the relative extrema of f? B.On what interval(s) is f ‘ constant? (-10, 0) C.On what interval(s) is f ‘ > 0? D.For what value(s) of x is f ‘ undefined? x = -10, x = 0, x = 3 x = -10, x = 3 This is the graph of f(x) on [-10, 3].

A.Where are the relative extrema of f? B.On what interval(s) is f ‘ constant? none C.On what interval(s) is f ‘ > 0? D.For what value(s) of x is f ‘ undefined? none x = -10, x = -1, x = 3 This is the graph of f ‘ (x) on [-10, 3].

Based upon the graph of f ‘ (x) given on the interval [0, 2pi], answer the following: Where does f achieve a minimum value? Round your answer(s) to three decimal places. x = 3.665, x = Where does f achieve a maximum value? Round your answer(s) to three decimal places. x = 0, x = CALCULATOR REQUIRED

Estimate to one decimal place the critical numbers of f. Estimate to one decimal place the value(s) of x at which there is a relative maximum. -1.4, 0.4 Given the graph of f(x) on to the right, answer the two questions below. -1.4, -0.4, 0.4, 1.6

Estimate to one decimal place the critical numbers of f. Estimate to one decimal place the value(s) of x at which there is a relative maximum. 1.1 Given the graph of f ‘ (x) on to the right, answer the three questions below. -1.9, 1.1, 1.8 Estimate to one decimal place the value(s) of x at which there is a relative minimum. -1.9, 1.8

a) For what value(s) of x will f have a horizontal tangent? 1 b) On what interval(s) will f be increasing? c) For what value(s) of x will f have a relative minimum? 1 d) For what value(s) of x will f have a relative maximum? none CALCULATOR REQUIRED

For what value(s) of x is f ‘ (x) = 0? On what interval(s) is f increasing?. Where are the relative maxima of f? -1 and 2 x = -1, x = 4 (-3, -1), (2, 4) This is the graph of f(x) on [-3, 4].

For what value(s) of x if f ‘ (x) = 0? For what value(s) of x does a relative maximum of f exist? On what interval(s) is f increasing? On what interval(s) is f concave up? -2, 1 and 3 -3, 1, 4 (-2, 1), (3, 4] (-3, -1), (2, 4) This is the graph of f ‘ (x) [-3, 4]

This is the graph of f(x) on [-5, 3] For what values of x if f undefined? On what interval(s) is f increasing? On what interval(s) is f ‘ < 0? Find the maximum value of f. 6 (-5, 1) (1, 3) -5, 1, 3

This is the graph of f ‘ (x) on [-7, 7]. For what value(s) of x is f ‘ (x) undefined? For what values of x is f ‘ > 0? On what interval(s) is f decreasing? On what interval(s) is f concave up? (0, 7) (-7, 0) (0, 7] none

This is the graph of f(x) on [-2, 2]. For what value(s) of x is f ‘ (x) = 0? For what value(s) of x does a relative minimum exist? On what interval(s) is f ‘ > 0? On what interval(s) is f “ > 0? (-1, 0), (1, 2) (-2, -1.5), (-0.5, 0.5), (1.5, 2) -2, -0.5, , -0.5, 0.5, 1.5

This is the graph of f ‘ (x) on [-2, 2] For what value(s) of x is f ‘ (x) = 0? For what value(s) of x is there a local minimum? On what interval(s) is f ‘ > 0? On what interval(s) is f “ > 0? (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-2, -1), (0, 1) -2, 0, 2 -2, -1, 0, 1, 2